Search Results (6 total)

A different approach will be presented that aims to scrutinize the phase-space trajectories of a general class of Hamiltonian systems with regard to their regular or irregular behavior. The approach is based on the “energy-second-moment map” that can be constructed for all Hamiltonian systems of the generic form H=p²∕2+V(q,t). With a three-component vector s consisting of the system’s energy h and second moments qp, q², this map linearly relates the vector s(t) at time t with the vector’s initial state s(0) at t=0. It will turn out that this map is directly obtained from the solution of a linear third-order equation that establishes an extension of the set of canonical equations. The Lyapunov functions of the energy-second-moment map will be shown to have simple analytical representations in terms of the solutions of this linear third-order equation. Applying Lyapunov’s regularity analysis for linear systems, we will show that the Lyapunov functions of the energy-second-moment map yields information on the irregularity of the particular phase-space trajectory. Our results will be illustrated by means of numerical examples.

Noether and Lie symmetry analyses based on point transformations that depend on time and spatial coordinates will be reviewed for a general class of time-dependent Hamiltonian systems. The resulting symmetries are expressed in the form of generators whose time-dependent coefficients follow as solutions of sets of ordinary differential (“auxiliary”) equations. The interrelation between the Noether and Lie sets of auxiliary equations will be elucidated. The auxiliary equations of the Noether approach will be shown to admit invariants for a much broader class of potentials, compared to earlier studies. As an example, we work out the Noether and Lie symmetries for the time-dependent Kepler system. The Runge-Lenz vector of the time-independent Kepler system will be shown to emerge as a Noether invariant if we adequately interpret the pertaining auxiliary equation. Furthermore, additional nonlocal invariants and symmetries of the Kepler system will be isolated by identifying further solutions of the auxiliary equations that depend on the explicitly known solution path of the equations of motion. Showing that the invariants remain unchanged under the action of different symmetry operators, we demonstrate that a unique correlation between a symmetry transformation and an invariant does not exist.

An exact invariant is derived for n-degree-of-freedom Hamiltonian systems with general time-dependent potentials. The invariant is worked out in two equivalent ways. In the first approach, we define a special Ansatz for the invariant and determine its time-dependent coefficients. In the second approach, we perform a two-step canonical transformation of the initially time-dependent Hamiltonian to a time-independent one. The invariant is found to contain a function of time f₂(t), defined as a solution of a linear third-order differential equation whose coefficients depend in general on the explicitly known configuration space trajectory that follows from the system’s time evolution. It is shown that the invariant can be interpreted as the time integral of an energy balance equation. Our result is applied to a one-dimensional, time-dependent, damped non-linear oscillator, and to a three-dimensional system of Coulomb-interacting particles that are confined in a time-dependent quadratic external potential. We finally show that our results can be used to assess the accuracy of numerical simulations of time-dependent Hamiltonian systems.

An exact invariant is derived for three-dimensional Hamiltonian systems of N particles confined within a general velocity-independent potential. The invariant is found to contain a time-dependent function f₂(t), embodying a solution of a third-order differential equation whose coefficients depend on the explicitly known trajectories of the particle ensemble. Our result is applied to a one-dimensional time-dependent nonlinear oscillator and to a system of Coulomb interacting particles in a time-dependent quadratic external potential.

The Vlasov equation embodies the smooth field approximation of the self-consistent equation of motion for charged particle beams. This framework is fundamentally altered if we include the fluctuating forces that originate from the actual charge granularity. We thereby perform the transition from a reversible description to a statistical mechanics description covering also the irreversible aspects of beam dynamics. Taking into account contributions from fluctuating forces is mandatory if we want to describe effects such as intrabeam scattering or temperature balancing within beams. Furthermore, the appearance of “discreteness errors” in computer simulations of beams can be modeled as “exact” beam dynamics that are being modified by fluctuating “error forces.” It will be shown that the related emittance increase depends on two distinct quantities: the magnitude of the fluctuating forces embodied in a friction coefficient, γ, and the correlation time dependent average temperature anisotropy. These analytical results are verified by various computer simulations.

Stochastic phenomena occurring within charged particle beams can be handled using the Vlasov-Fokker-Planck generalization of the Vlasov equation. In particular, this nondeterministic approach can deal with effects due to Coulomb scattering between the beam particles. Moreover, stochastic phenomena also occur in computer simulations of charged particle beams. Both processes—although different in their physical nature—can be described by the Vlasov-Fokker-Planck equation, since in both cases the underlying stochastic process can be classified as a Markov process. This description is applied to beams in periodic focusing systems. We derive an equation relating the change of the μ-phase space entropy to the change of rms emittance and ‘‘temperature weighted excess field energy.’’ This equation enables us both to improve our capability to interpret the results of computer simulations, as well as to identify the conditions needed to minimize scattering induced degradation of the quality of beams circulating in storage rings. © 1996 The American Physical Society.